Rise time

In electronics, when describing a voltage or currentstep function, rise time is the time taken by a signal to change from a specified low value to a specified high value. Typically, in analog electronics, these values are 10% and 90% of the step height: in control theory applications, according to Levine (1996, p. 158), rise time is defined as "the time required for the response to rise from x% to y% of its final value", with 0%-100% rise time common for underdamped second order systems, 5%-95% for critically damped and 10%-90% for overdamped.[1] The output signal of a system is characterized also by fall time: both parameters depend on rise and fall times of input signal and on the characteristics of the system.

Rise time is an analog parameter of fundamental importance in high speed electronics, since it is a measure of the ability of a circuit to respond to fast input signals. Many efforts over the years have been made to reduce the rise times of generators, analog and digital circuits, measuring and data transmission equipment, focused on the research of faster electron devices and on techniques of reduction of stray circuit parameters (mainly capacitances and inductances). For applications outside the realm of high speed electronics, long (compared to the attainable state of the art) rise times are sometimes desirable: examples are the dimming of a light, where a longer rise-time results, amongst other things, in a longer life for the bulb, or digital signals apt to the control of analog ones, where a longer rise time means lower capacitive feedthrough, and thus lower coupling noise.

For a simple one-stage low-pass RC network, also known as a single-pole filter, the 10% to 90% rise time is proportional to the network time constant :

The proportionality constant can be derived by using the output response of the network to a unit step function input signal of amplitude, also known as its step response:

Solving for time

We call t1 the time needed to go from 0% to 10% of the steady-state value, and t2 the one to 90%. Thus t1 is such that and t2 is such that . Solving the previous equation for these two values we find the analytical expression for t1 and t2:

We obtain t2 in the same way, resulting in

Subtracting from we obtain the rise time, which is therefore proportional to the time constant:

Consider a system composed by cascaded non interacting blocks, each having a rise time and no overshoot in their step response: suppose also that the input signal of the first block has a rise time whose value is . Then its output signal has a rise time equal to

^This beautiful one-page paper does not contain any calculation. Henry Wallman simply sets up a table he calls dictionary paralleling concepts from electronics engineering and probability theory: the key of the process is the use of Laplace transform. Then he notes that, following the correspondence of concepts established by the dictionary, that the step response of a cascade of blocks corresponds to the central limit theorem and states that: "This has important practical consequences, among them the fact that if a network is free of overshoot its time-of-response inevitably increases rapidly upon cascading, namely as the square-root of the number of cascaded network".(Wallman 1950, p. 91)